Rigidity of a Non-elliptic Differential Inclusion Related to the Aviles–Giga Conjecture

Lamy, Xavier; Lorent, Andrew; Peng, Guanying

doi:10.1007/s00205-020-01545-z

Cited by 14 publications

(20 citation statements)

References 34 publications

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“…This estimate is of exactly the same type as the estimate (51) in [LLP20], which is shown to lead to a contradiction because of well known properties of the Hilbert transform; see the arguments below equation (51) in [LLP20] for the details. The contradiction from the estimate (109) shows that ie i2θ(x) • div Σ (m) (x) = 0 for all x ∈ G, and thus establishes (104).…”

Section: Factorization For General Entropiesmentioning

confidence: 65%

“…The rest of the game involves showing ie i2θ •div Σ(m) = 0 using a similar contradiction argument as in [LLP20]. Specifically, assuming ie i2θ • div Σ(m) = 0 at a generic point x, then estimates established for the coefficients A Φ j in Lemma 27 plugged into (16) would force A Φ 2 to satisfy a bound of the form…”

Section: Ideas In the Proof Of Theoremmentioning

confidence: 99%

“…The proof of Proposition 13 helps strengthen an earlier version of our Theorem 3. Also note that Proposition 30 (including Lemmas 34 and 35) was originally in the first posted version of [LLP20] and as such is joint work of the three of us. A. L. gratefully acknowledges the support of the Simons foundation, collaboration grant #426900.…”

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confidence: 98%

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Factorization for entropy production of the Eikonal equation and regularity

Lorent,

Peng

2021

Preprint

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The Eikonal equation arises naturally in the limit of the second order Aviles-Giga functional whose Γ-convergence is a long standing challenging problem. The theory of entropy solutions of the Eikonal equation plays a central role in the variational analysis of this problem. Establishing fine structures of entropy solutions of the Eikonal equation, e.g. concentration of entropy measures on H 1 -rectifiable sets in 2D, is arguably the key missing part for a proof of the full Γ-convergence of the Aviles-Giga functional. In the first part of this work, for p ∈ 1, 4 3 we establish an L p version of the main theorem of [GL20]. Specifically we show that if m is a solution to the Eikonal equation, then m ∈ B 1 3 3p,∞,loc is equivalent to all entropy productions of m being in L p loc . Given the main result of [GL20], this result also shows that as a consequence of a weak form of the Aviles-Giga conjecture (namely the conjecture that all solutions to the Eikonal equation whose entropy productions are in L p loc are rigid) -the rigidity/flexibility threshold of the Eikonal equation is exactly the space B 1 3 3,∞,loc . In the second part of this paper, under the assumption that all entropy productions are in L p loc , we establish a factorization formula for entropy productions of solutions of the Eikonal equation in terms of the two Jin-Kohn entropies. A consequence of this formula is control of all entropy productions by the Jin-Kohn entropies in the L p setting -this is a strong extension of the main result of [LP18].

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Section: Factorization For General Entropiesmentioning

confidence: 65%

Section: Ideas In the Proof Of Theoremmentioning

confidence: 99%

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Factorization for entropy production of the Eikonal equation and regularity

Lorent,

Peng

2021

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“…u(x) = ± x/|x| locally, up to an origin-shift). Recently, [LP18,LLP20] proved that the same conclusion can be obtained under the weaker assumption that the entropy production coming from the Jin-Kohn entropies vanishes. An important further step was obtained in [DLO03] (see also [AKLR02]), where it is shown that configurations of finite energy share some of the characteristic properties of BV -mappings.…”

Section: Introductionmentioning

confidence: 58%

“…Indeed, that proof is almost entirely based on the use of the entropies Φ ξ defined in (1.16) in order to follow in a weak sense the characteristics of the eikonal equation (which are lines). In the unoriented case, we build instead on a method of Lorent and Peng [LP18, Theorem 4] (see also [LLP20,Lemma 7]) which in turn is inspired by the earlier work [ Š93] of Šverák on differential inclusions. We apply it in Proposition 6.1 with the trigonometric entropies to show that if v is a zero-state then v ∈ W 1,3/2 loc (Ω) (actually W 1,p loc for every p < 2, see Remark 6.2) and for every open set ω ⊂⊂ Ω and |h| 1,…”

Section: Complex Representationmentioning

confidence: 99%

Compactness and structure of zero-states for unoriented Aviles-Giga functionals

Goldman,

Merlet,

Pegon

et al. 2021

Preprint

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Motivated by some models of pattern formation involving an unoriented director field in the plane, we study a family of unoriented counterparts to the Aviles-Giga functional.We introduce a nonlinear curl operator for such unoriented vector fields as well as a family of even entropies which we call "trigonometric entropies". Using these tools we show two main theorems which parallel some results in the literature on the classical Aviles-Giga energy. The first is a compactness result for sequences of configurations with uniformly bounded energies. The second is a complete characterization of zero-states, that is, the limit configurations when the energies go to 0. These are Lipschitz continuous away from a locally finite set of points, near which they form either a vortex pattern or a disclination with degree 1/2. The proof is based on a combination of regularity theory together with techniques coming from the study of the Ginzburg-Landau energy. Our methods provide alternative proofs in the classical Aviles-Giga context.

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A Model Problem for Nematic-Isotropic Transitions with Highly Disparate Elastic Constants

Golovaty

Novack

Sternberg

et al. 2020

Arch Rational Mech Anal

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Continuing the program initiated in [17], we analyze a model problem based on highly disparate elastic constants that we propose in order to understand corners and cusps that form on the boundary between the nematic and isotropic phases in a liquid crystal. For a bounded planar domain Ω we investigate the ε → 0 asymptotics of the variational problemwithin various parameter regimes for L ε > 0. Here u : Ω → R 2 and W is a potential vanishing on the unit circle and at the origin. When ε L ε → 0, we show that these functionals Γ−converge to a constant multiple of the perimeter of the phase boundary and the divergence penalty is not felt. However, when L ε ≡ L > 0, we find that a tangency requirement along the phase boundary for competitors in the conjectured Γ-limit becomes a mechanism for development of singularities. We establish criticality conditions for this limit and under a non-degeneracy assumption on the potential we prove compactness of energy bounded sequences in L 2 . The role played by this tangency condition on the formation of interfacial singularities is investigated through several examples: each of these examples involves analytically rigorous reasoning motivated by numerical experiments. We argue that generically, "wall" singularities between S 1 -valued states of the kind analyzed in [17] are expected near the defects along the phase boundary. *

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Rigidity of a Non-elliptic Differential Inclusion Related to the Aviles–Giga Conjecture

Cited by 14 publications

References 34 publications

Factorization for entropy production of the Eikonal equation and regularity

Factorization for entropy production of the Eikonal equation and regularity

Compactness and structure of zero-states for unoriented Aviles-Giga functionals

A Model Problem for Nematic-Isotropic Transitions with Highly Disparate Elastic Constants

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